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Extra info for Social Choice and the Mathematics of Manipulation (Outlooks)

Example text

It is assumed further that the choice is made in this way: Before knowing the set S, the chooser considers in turn all possible pairs of alternatives, say x and y, and for each pair he makes one and only one of three decisions: x is preferred to y, x is indifferent to y, or y is preferred to x. The decisions made for different pairs are assumed to be consistent with one another, so that, for example, if x is preferred to y and y to z, then x is preferred to z; similarly, if x is indifferent to y and y to z, then x is indifferent to z.

6) [C] Use the following profile P to show that the narrow Borda rule can differ from the broad Borda rule. That is, find a subset v of alternatives such that, if V is the Borda rule, then V(P|v) is not the alternative in v with the highest Borda score from P. P d a b c b c d a c d a b d a c b The following two exercises outline a derivation showing that the version of Arrow’s theorem for voting rules implies the version for social welfare functions (and thus, because of transitive rationality, the version for social choice functions).

PIIA) Pairwise Independence of Irrelevant Alternatives: For every two (A, n)-profiles P and P , if V(P) = R and V(P ) = R , then for every pair of alternatives x, y ∈ A, if Ri |{x, y} = Ri {x, y} for every i, then R|{x, y} = R |{x, y}. 6 The version of IIA we use is inspired by the pairwise version of IIA occurring in Arrow’s theorem for social welfare functions. Intuitively, that pairwise version says that society’s preference for x over y should be preserved as long as no voter changes the order in which he or 6 The first version of Arrow’s theorem in the context of voting rules seems to be due to Hansson (1969).